If a mixed number be given, reduce it to an improper fraction, and then find its logarithm as above. Ans.-1.9488475. 23. What is the logarithm of? 24. What is the logarithm of 124? Ans. 1.0969100. 36. PROPORTION BY LOGARITHMS. RULE I. Prepare the terms as in the Rule of Three of Decimals, if they require it. II. Place the terms (so reduced) in order, one under another, with the logarithm of each opposite to its respective term. III. Add the logarithms of the two multiplying terms together, and from the sum subtract the logarithm of the dividing term; the remainder will be the logarithm of the answer. IV. Or find the arithmetical complement of the dividing term, and add it and the logarithms of the two remaining terms together, observing to take the difference of 10 and the index, as in division; the result will be the logarithm of the answer as before. EXAMPLES. 1. If the week's allowance for 5 seamen be 381b. of biscuit, how many pounds will a ship's company of 224 men consume in the same time? VOL. I. Having stated the question according to the rule, I find that it belongs to the direct rule, and therefore the first is the dividing term; whence, under the first method, I add the logarithms of the first and second terms together, and subtract that of the first from the sum. By the second method, I find the arithmetical complement of the first term, and the logarithms of the two other terms, and add all the three together, subtracting 10 from the index of the sum; the result by both methods is the same, namely, the logarithm of the answer, which is of the same name with the second term, viz. pounds. 2. If 7 men can perform a piece of work in 54 days, in how many days would 23 men accomplish the same? As 7 men. To 54 days. First method. ..... ... OPERATION. log. 0.8450980 | Log. So are 23 men. .log. = 1.3617278 • ... Log. Second method. =0.8450980 = 1.7323938: Arith. co. 8.6382722 Sum 1.2157640 This example evidently belongs to the inverse rule, I therefore add the logs. of the first and second terms together; and from the sum subtract the log. of the third, according to the 1st method, or add its arithmetical complement, according to the 2nd. 3. If 34lb. of tea cost 11. 7s. 6d. what is the value of 2cwt. 3qr. 4lb. ? Explanation. The first term, and the qrs. and lbs. in the third, are reduced to decimals of a cwt.; and the shillings and pence in the second, to the decimal of a pound; after which the operation proceeds exactly as before. 4. If 3 yards of lace sell for 19s. 6d.+, what is the value of 17 pieces, each 16yds. 3qr. 2na. ? Explanation. This operation at first sight appears to have four terms, but the two latter evidently constitute but one term, namely, the third; the pieces and yards are multiplied together by adding their logarithms. 5. Find a fourth proportional to 123.4... 43.21, and 1. As 123.4... To 43.21 log. =1.6355843 .... .log. =2.0913152 So is 1 ..... log. = 0.0000000 ...= 0.0000000 6. If a pipe of Madeira cost 1097. 10s. what must be given for 1hhd. 25 gallons of the same? So are 1hhd. 25 gal. = .69841 pipe = - 1.8441104 To 76.4761. = 761. 9s. 6d. . = 1.8835245 7. If 20lb. of sugar cost 11. 2s. 6d. what will 56ĺb. cost? Answer 31. 3s. 8. Required a fourth proportional to the three given numbers, 11, 12, and 20? Ans. 21.818, &c. 10. A person receives 821. 7s. 6d. per year, how much is that per day? Ans. 4s. 6d. 11. Required a third proportional to 123 and 234? Answer 445.17, &c. 12. If A lends B 3321. for 7 months, what sum ought B to lend A for 20 months, to discharge the obligation? Answer 1161. 48. 37. INVOLUTION BY LOGARITHMS. RULE I. Multiply the logarithm of the number to be involved, by the index of the proposed power, and the product will be the logarithm of the power. II. If the index of the logarithm be negative, the product of the index will be negative; and since what is carried is affirmative, the difference of these two must be taken, and the sign of greater prefixed to it, for the index of the logarithm of the proposed power. |